LMFDB - Embedded newform 5520.2.be.c.1471.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1471,5520)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5520.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.be (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$44.0774219157$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.4
Character	$\chi$	$=$	5520.1471
Dual form			5520.2.be.c.1471.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000i q^{5} -1.52302 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +1.00000i q^{5} -1.52302 q^{7} -1.00000 q^{9} -6.43261 q^{11} -4.50229 q^{13} -1.00000 q^{15} +1.65230i q^{17} +5.57593 q^{19} -1.52302i q^{21} +(2.99135 - 3.74857i) q^{23} -1.00000 q^{25} -1.00000i q^{27} -2.30375 q^{29} -1.77291i q^{31} -6.43261i q^{33} -1.52302i q^{35} +5.43051i q^{37} -4.50229i q^{39} -2.07053 q^{41} -4.75591 q^{43} -1.00000i q^{45} -2.99382i q^{47} -4.68040 q^{49} -1.65230 q^{51} -2.72374i q^{53} -6.43261i q^{55} +5.57593i q^{57} -3.99055i q^{59} +9.74609i q^{61} +1.52302 q^{63} -4.50229i q^{65} +3.71642 q^{67} +(3.74857 + 2.99135i) q^{69} -2.39857i q^{71} -9.23774 q^{73} -1.00000i q^{75} +9.79702 q^{77} +11.4088 q^{79} +1.00000 q^{81} +9.35972 q^{83} -1.65230 q^{85} -2.30375i q^{87} -2.66224i q^{89} +6.85710 q^{91} +1.77291 q^{93} +5.57593i q^{95} -8.66759i q^{97} +6.43261 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 8 q^{7} - 32 q^{9}+O(q^{10}) $ 32 * q - 8 * q^7 - 32 * q^9	$ 32 q - 8 q^{7} - 32 q^{9} + 8 q^{11} - 8 q^{13} - 32 q^{15} - 32 q^{25} + 4 q^{29} + 20 q^{41} + 52 q^{49} - 4 q^{51} + 8 q^{63} + 32 q^{67} - 40 q^{73} - 24 q^{77} + 32 q^{79} + 32 q^{81} - 4 q^{85} - 48 q^{91} - 8 q^{99}+O(q^{100}) $ 32 * q - 8 * q^7 - 32 * q^9 + 8 * q^11 - 8 * q^13 - 32 * q^15 - 32 * q^25 + 4 * q^29 + 20 * q^41 + 52 * q^49 - 4 * q^51 + 8 * q^63 + 32 * q^67 - 40 * q^73 - 24 * q^77 + 32 * q^79 + 32 * q^81 - 4 * q^85 - 48 * q^91 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times$.

$n$	$1201$	$1381$	$1841$	$4417$	$4831$
$\chi(n)$	$-1$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	−1.52302			−0.575649			−0.287824	−	0.957683i	$-0.592932\pi$
$7$	−1.52302			−0.575649			−0.287824	+	0.957683i	$0.592932\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−6.43261			−1.93950			−0.969752	−	0.244092i	$-0.921510\pi$
$11$	−6.43261			−1.93950			−0.969752	+	0.244092i	$0.921510\pi$
$12$		0			0
$13$	−4.50229			−1.24871			−0.624356	−	0.781140i	$-0.714639\pi$
$13$	−4.50229			−1.24871			−0.624356	+	0.781140i	$0.714639\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$			1.65230i			0.400743i	0.979720	+	0.200371i	$0.0642148\pi$
$17$			1.65230i			0.400743i	−0.979720	+	0.200371i	$0.935785\pi$
$18$		0			0
$19$	5.57593			1.27921			0.639603	−	0.768705i	$-0.279099\pi$
$19$	5.57593			1.27921			0.639603	+	0.768705i	$0.279099\pi$
$20$		0			0
$21$		−	1.52302i		−	0.332351i
$22$		0			0
$23$	2.99135	−	3.74857i	0.623740	−	0.781632i
$23$	2.99135	−	3.74857i	0.623740	−	0.781632i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−2.30375			−0.427796			−0.213898	−	0.976856i	$-0.568616\pi$
$29$	−2.30375			−0.427796			−0.213898	+	0.976856i	$0.568616\pi$
$30$		0			0
$31$		−	1.77291i		−	0.318424i	−0.987244	−	0.159212i	$-0.949105\pi$
$31$		−	1.77291i		−	0.318424i	0.987244	−	0.159212i	$-0.0508953\pi$
$32$		0			0
$33$		−	6.43261i		−	1.11977i
$34$		0			0
$35$		−	1.52302i		−	0.257438i
$36$		0			0
$37$			5.43051i			0.892771i	0.894841	+	0.446385i	$0.147289\pi$
$37$			5.43051i			0.892771i	−0.894841	+	0.446385i	$0.852711\pi$
$38$		0			0
$39$		−	4.50229i		−	0.720944i
$40$		0			0
$41$	−2.07053			−0.323362			−0.161681	−	0.986843i	$-0.551692\pi$
$41$	−2.07053			−0.323362			−0.161681	+	0.986843i	$0.551692\pi$
$42$		0			0
$43$	−4.75591			−0.725270			−0.362635	−	0.931931i	$-0.618123\pi$
$43$	−4.75591			−0.725270			−0.362635	+	0.931931i	$0.618123\pi$
$44$		0			0
$45$		−	1.00000i		−	0.149071i
$46$		0			0
$47$		−	2.99382i		−	0.436693i	−0.975871	−	0.218347i	$-0.929934\pi$
$47$		−	2.99382i		−	0.436693i	0.975871	−	0.218347i	$-0.0700663\pi$
$48$		0			0
$49$	−4.68040			−0.668628
$50$		0			0
$51$	−1.65230			−0.231369
$52$		0			0
$53$		−	2.72374i		−	0.374134i	−0.982347	−	0.187067i	$-0.940102\pi$
$53$		−	2.72374i		−	0.374134i	0.982347	−	0.187067i	$-0.0598982\pi$
$54$		0			0
$55$		−	6.43261i		−	0.867373i
$56$		0			0
$57$			5.57593i			0.738550i
$58$		0			0
$59$		−	3.99055i		−	0.519525i	−0.965673	−	0.259763i	$-0.916356\pi$
$59$		−	3.99055i		−	0.519525i	0.965673	−	0.259763i	$-0.0836443\pi$
$60$		0			0
$61$			9.74609i			1.24786i	0.781480	+	0.623930i	$0.214465\pi$
$61$			9.74609i			1.24786i	−0.781480	+	0.623930i	$0.785535\pi$
$62$		0			0
$63$	1.52302			0.191883
$64$		0			0
$65$		−	4.50229i		−	0.558441i
$66$		0			0
$67$	3.71642			0.454033			0.227017	−	0.973891i	$-0.427103\pi$
$67$	3.71642			0.454033			0.227017	+	0.973891i	$0.427103\pi$
$68$		0			0
$69$	3.74857	+	2.99135i	0.451275	+	0.360117i
$70$		0			0
$71$		−	2.39857i		−	0.284658i	−0.989819	−	0.142329i	$-0.954541\pi$
$71$		−	2.39857i		−	0.284658i	0.989819	−	0.142329i	$-0.0454591\pi$
$72$		0			0
$73$	−9.23774			−1.08120			−0.540598	−	0.841281i	$-0.681802\pi$
$73$	−9.23774			−1.08120			−0.540598	+	0.841281i	$0.681802\pi$
$74$		0			0
$75$		−	1.00000i		−	0.115470i
$76$		0			0
$77$	9.79702			1.11647
$78$		0			0
$79$	11.4088			1.28360			0.641798	−	0.766874i	$-0.278189\pi$
$79$	11.4088			1.28360			0.641798	+	0.766874i	$0.278189\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	9.35972			1.02736			0.513681	−	0.857981i	$-0.328281\pi$
$83$	9.35972			1.02736			0.513681	+	0.857981i	$0.328281\pi$
$84$		0			0
$85$	−1.65230			−0.179218
$86$		0			0
$87$		−	2.30375i		−	0.246988i
$88$		0			0
$89$		−	2.66224i		−	0.282197i	−0.989996	−	0.141099i	$-0.954937\pi$
$89$		−	2.66224i		−	0.282197i	0.989996	−	0.141099i	$-0.0450634\pi$
$90$		0			0
$91$	6.85710			0.718820
$92$		0			0
$93$	1.77291			0.183842
$94$		0			0
$95$			5.57593i			0.572078i
$96$		0			0
$97$		−	8.66759i		−	0.880060i	−0.897983	−	0.440030i	$-0.854968\pi$
$97$		−	8.66759i		−	0.880060i	0.897983	−	0.440030i	$-0.145032\pi$
$98$		0			0
$99$	6.43261			0.646501
$100$		0			0
$101$	18.6952			1.86024			0.930121	−	0.367254i	$-0.119702\pi$
$101$	18.6952			1.86024			0.930121	+	0.367254i	$0.119702\pi$
$102$		0			0
$103$	7.63952			0.752744			0.376372	−	0.926469i	$-0.377171\pi$
$103$	7.63952			0.752744			0.376372	+	0.926469i	$0.377171\pi$
$104$		0			0
$105$	1.52302			0.148632
$106$		0			0
$107$	6.71244			0.648916			0.324458	−	0.945900i	$-0.394818\pi$
$107$	6.71244			0.648916			0.324458	+	0.945900i	$0.394818\pi$
$108$		0			0
$109$		−	2.70608i		−	0.259196i	−0.991567	−	0.129598i	$-0.958631\pi$
$109$		−	2.70608i		−	0.259196i	0.991567	−	0.129598i	$-0.0413687\pi$
$110$		0			0
$111$	−5.43051			−0.515441
$112$		0			0
$113$			0.212216i			0.0199636i	0.999950	+	0.00998181i	$0.00317736\pi$
$113$			0.212216i			0.0199636i	−0.999950	+	0.00998181i	$0.996823\pi$
$114$		0			0
$115$	3.74857	+	2.99135i	0.349556	+	0.278945i
$116$		0			0
$117$	4.50229			0.416237
$118$		0			0
$119$		−	2.51650i		−	0.230687i
$120$		0			0
$121$	30.3784			2.76168
$122$		0			0
$123$		−	2.07053i		−	0.186693i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			16.8377i			1.49410i	0.664767	+	0.747051i	$0.268531\pi$
$127$			16.8377i			1.49410i	−0.664767	+	0.747051i	$0.731469\pi$
$128$		0			0
$129$		−	4.75591i		−	0.418735i
$130$		0			0
$131$		−	11.1242i		−	0.971929i	−0.873979	−	0.485964i	$-0.838468\pi$
$131$		−	11.1242i		−	0.971929i	0.873979	−	0.485964i	$-0.161532\pi$
$132$		0			0
$133$	−8.49227			−0.736374
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$			5.56165i			0.475164i	0.971368	+	0.237582i	$0.0763548\pi$
$137$			5.56165i			0.475164i	−0.971368	+	0.237582i	$0.923645\pi$
$138$		0			0
$139$		−	5.63364i		−	0.477839i	−0.971039	−	0.238920i	$-0.923207\pi$
$139$		−	5.63364i		−	0.477839i	0.971039	−	0.238920i	$-0.0767932\pi$
$140$		0			0
$141$	2.99382			0.252125
$142$		0			0
$143$	28.9615			2.42188
$144$		0			0
$145$		−	2.30375i		−	0.191316i
$146$		0			0
$147$		−	4.68040i		−	0.386033i
$148$		0			0
$149$		−	10.5221i		−	0.862001i	−0.902352	−	0.431000i	$-0.858161\pi$
$149$		−	10.5221i		−	0.862001i	0.902352	−	0.431000i	$-0.141839\pi$
$150$		0			0
$151$		−	17.5034i		−	1.42441i	−0.701972	−	0.712205i	$-0.747697\pi$
$151$		−	17.5034i		−	1.42441i	0.701972	−	0.712205i	$-0.252303\pi$
$152$		0			0
$153$		−	1.65230i		−	0.133581i
$154$		0			0
$155$	1.77291			0.142404
$156$		0			0
$157$			7.31873i			0.584098i	0.956403	+	0.292049i	$0.0943371\pi$
$157$			7.31873i			0.584098i	−0.956403	+	0.292049i	$0.905663\pi$
$158$		0			0
$159$	2.72374			0.216007
$160$		0			0
$161$	−4.55590	+	5.70917i	−0.359055	+	0.449946i
$162$		0			0
$163$			9.06133i			0.709738i	0.934916	+	0.354869i	$0.115475\pi$
$163$			9.06133i			0.709738i	−0.934916	+	0.354869i	$0.884525\pi$
$164$		0			0
$165$	6.43261			0.500778
$166$		0			0
$167$		−	8.02942i		−	0.621335i	−0.950519	−	0.310667i	$-0.899447\pi$
$167$		−	8.02942i		−	0.621335i	0.950519	−	0.310667i	$-0.100553\pi$
$168$		0			0
$169$	7.27066			0.559281
$170$		0			0
$171$	−5.57593			−0.426402
$172$		0			0
$173$	−0.927933			−0.0705495			−0.0352747	−	0.999378i	$-0.511231\pi$
$173$	−0.927933			−0.0705495			−0.0352747	+	0.999378i	$0.511231\pi$
$174$		0			0
$175$	1.52302			0.115130
$176$		0			0
$177$	3.99055			0.299948
$178$		0			0
$179$			25.1922i			1.88295i	0.337082	+	0.941475i	$0.390560\pi$
$179$			25.1922i			1.88295i	−0.337082	+	0.941475i	$0.609440\pi$
$180$		0			0
$181$			1.69810i			0.126219i	0.998007	+	0.0631095i	$0.0201017\pi$
$181$			1.69810i			0.126219i	−0.998007	+	0.0631095i	$0.979898\pi$
$182$		0			0
$183$	−9.74609			−0.720452
$184$		0			0
$185$	−5.43051			−0.399259
$186$		0			0
$187$		−	10.6286i		−	0.777242i
$188$		0			0
$189$			1.52302i			0.110784i
$190$		0			0
$191$	13.0724			0.945886			0.472943	−	0.881093i	$-0.343192\pi$
$191$	13.0724			0.945886			0.472943	+	0.881093i	$0.343192\pi$
$192$		0			0
$193$	16.0209			1.15321			0.576606	−	0.817023i	$-0.304377\pi$
$193$	16.0209			1.15321			0.576606	+	0.817023i	$0.304377\pi$
$194$		0			0
$195$	4.50229			0.322416
$196$		0			0
$197$	−11.9113			−0.848647			−0.424323	−	0.905511i	$-0.639488\pi$
$197$	−11.9113			−0.848647			−0.424323	+	0.905511i	$0.639488\pi$
$198$		0			0
$199$	−7.54875			−0.535117			−0.267558	−	0.963542i	$-0.586217\pi$
$199$	−7.54875			−0.535117			−0.267558	+	0.963542i	$0.586217\pi$
$200$		0			0
$201$			3.71642i			0.262136i
$202$		0			0
$203$	3.50867			0.246261
$204$		0			0
$205$		−	2.07053i		−	0.144612i
$206$		0			0
$207$	−2.99135	+	3.74857i	−0.207913	+	0.260544i
$208$		0			0
$209$	−35.8678			−2.48103
$210$		0			0
$211$			4.69539i			0.323244i	0.986853	+	0.161622i	$0.0516726\pi$
$211$			4.69539i			0.323244i	−0.986853	+	0.161622i	$0.948327\pi$
$212$		0			0
$213$	2.39857			0.164347
$214$		0			0
$215$		−	4.75591i		−	0.324351i
$216$		0			0
$217$			2.70018i			0.183300i
$218$		0			0
$219$		−	9.23774i		−	0.624229i
$220$		0			0
$221$		−	7.43916i		−	0.500412i
$222$		0			0
$223$		−	24.1553i		−	1.61756i	−0.588110	−	0.808781i	$-0.700128\pi$
$223$		−	24.1553i		−	1.61756i	0.588110	−	0.808781i	$-0.299872\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	12.4903			0.829010			0.414505	−	0.910047i	$-0.363955\pi$
$227$	12.4903			0.829010			0.414505	+	0.910047i	$0.363955\pi$
$228$		0			0
$229$			29.8509i			1.97260i	0.164951	+	0.986302i	$0.447253\pi$
$229$			29.8509i			1.97260i	−0.164951	+	0.986302i	$0.552747\pi$
$230$		0			0
$231$			9.79702i			0.644596i
$232$		0			0
$233$	−0.897082			−0.0587698			−0.0293849	−	0.999568i	$-0.509355\pi$
$233$	−0.897082			−0.0587698			−0.0293849	+	0.999568i	$0.509355\pi$
$234$		0			0
$235$	2.99382			0.195295
$236$		0			0
$237$			11.4088i			0.741084i
$238$		0			0
$239$		−	25.9763i		−	1.68027i	−0.542381	−	0.840133i	$-0.682477\pi$
$239$		−	25.9763i		−	1.68027i	0.542381	−	0.840133i	$-0.317523\pi$
$240$		0			0
$241$		−	8.15296i		−	0.525179i	−0.964908	−	0.262589i	$-0.915424\pi$
$241$		−	8.15296i		−	0.525179i	0.964908	−	0.262589i	$-0.0845764\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$		−	4.68040i		−	0.299020i
$246$		0			0
$247$	−25.1045			−1.59736
$248$		0			0
$249$			9.35972i			0.593148i
$250$		0			0
$251$	−6.31194			−0.398406			−0.199203	−	0.979958i	$-0.563835\pi$
$251$	−6.31194			−0.398406			−0.199203	+	0.979958i	$0.563835\pi$
$252$		0			0
$253$	−19.2422	+	24.1131i	−1.20975	+	1.51598i
$254$		0			0
$255$		−	1.65230i		−	0.103471i
$256$		0			0
$257$	−10.8718			−0.678164			−0.339082	−	0.940757i	$-0.610116\pi$
$257$	−10.8718			−0.678164			−0.339082	+	0.940757i	$0.610116\pi$
$258$		0			0
$259$		−	8.27080i		−	0.513923i
$260$		0			0
$261$	2.30375			0.142599
$262$		0			0
$263$	23.5415			1.45163			0.725814	−	0.687891i	$-0.241463\pi$
$263$	23.5415			1.45163			0.725814	+	0.687891i	$0.241463\pi$
$264$		0			0
$265$	2.72374			0.167318
$266$		0			0
$267$	2.66224			0.162927
$268$		0			0
$269$	13.5049			0.823408			0.411704	−	0.911318i	$-0.364934\pi$
$269$	13.5049			0.823408			0.411704	+	0.911318i	$0.364934\pi$
$270$		0			0
$271$			18.2034i			1.10578i	0.833255	+	0.552889i	$0.186475\pi$
$271$			18.2034i			1.10578i	−0.833255	+	0.552889i	$0.813525\pi$
$272$		0			0
$273$			6.85710i			0.415011i
$274$		0			0
$275$	6.43261			0.387901
$276$		0			0
$277$	−9.79569			−0.588566			−0.294283	−	0.955718i	$-0.595081\pi$
$277$	−9.79569			−0.588566			−0.294283	+	0.955718i	$0.595081\pi$
$278$		0			0
$279$			1.77291i			0.106141i
$280$		0			0
$281$		−	10.6935i		−	0.637923i	−0.947768	−	0.318962i	$-0.896666\pi$
$281$		−	10.6935i		−	0.637923i	0.947768	−	0.318962i	$-0.103334\pi$
$282$		0			0
$283$	−27.2186			−1.61797			−0.808987	−	0.587826i	$-0.799984\pi$
$283$	−27.2186			−1.61797			−0.808987	+	0.587826i	$0.799984\pi$
$284$		0			0
$285$	−5.57593			−0.330290
$286$		0			0
$287$	3.15346			0.186143
$288$		0			0
$289$	14.2699			0.839405
$290$		0			0
$291$	8.66759			0.508103
$292$		0			0
$293$			23.1769i			1.35401i	0.735979	+	0.677004i	$0.236722\pi$
$293$			23.1769i			1.35401i	−0.735979	+	0.677004i	$0.763278\pi$
$294$		0			0
$295$	3.99055			0.232339
$296$		0			0
$297$			6.43261i			0.373258i
$298$		0			0
$299$	−13.4680	+	16.8772i	−0.778872	+	0.976033i
$300$		0			0
$301$	7.24337			0.417501
$302$		0			0
$303$			18.6952i			1.07401i
$304$		0			0
$305$	−9.74609			−0.558060
$306$		0			0
$307$		−	23.1278i		−	1.31997i	−0.751278	−	0.659985i	$-0.770562\pi$
$307$		−	23.1278i		−	1.31997i	0.751278	−	0.659985i	$-0.229438\pi$
$308$		0			0
$309$			7.63952i			0.434597i
$310$		0			0
$311$		−	15.5382i		−	0.881091i	−0.897730	−	0.440545i	$-0.854785\pi$
$311$		−	15.5382i		−	0.881091i	0.897730	−	0.440545i	$-0.145215\pi$
$312$		0			0
$313$		−	1.93700i		−	0.109486i	−0.998500	−	0.0547428i	$-0.982566\pi$
$313$		−	1.93700i		−	0.109486i	0.998500	−	0.0547428i	$-0.0174339\pi$
$314$		0			0
$315$			1.52302i			0.0858127i
$316$		0			0
$317$	−20.0281			−1.12489			−0.562446	−	0.826834i	$-0.690140\pi$
$317$	−20.0281			−1.12489			−0.562446	+	0.826834i	$0.690140\pi$
$318$		0			0
$319$	14.8191			0.829713
$320$		0			0
$321$			6.71244i			0.374652i
$322$		0			0
$323$			9.21313i			0.512632i
$324$		0			0
$325$	4.50229			0.249742
$326$		0			0
$327$	2.70608			0.149647
$328$		0			0
$329$			4.55966i			0.251382i
$330$		0			0
$331$		−	19.6804i		−	1.08173i	−0.841109	−	0.540866i	$-0.818097\pi$
$331$		−	19.6804i		−	1.08173i	0.841109	−	0.540866i	$-0.181903\pi$
$332$		0			0
$333$		−	5.43051i		−	0.297590i
$334$		0			0
$335$			3.71642i			0.203050i
$336$		0			0
$337$			3.73958i			0.203708i	0.994799	+	0.101854i	$0.0324774\pi$
$337$			3.73958i			0.203708i	−0.994799	+	0.101854i	$0.967523\pi$
$338$		0			0
$339$	−0.212216			−0.0115260
$340$		0			0
$341$			11.4044i			0.617585i
$342$		0			0
$343$	17.7895			0.960544
$344$		0			0
$345$	−2.99135	+	3.74857i	−0.161049	+	0.201816i
$346$		0			0
$347$		−	32.8358i		−	1.76272i	−0.472449	−	0.881358i	$-0.656630\pi$
$347$		−	32.8358i		−	1.76272i	0.472449	−	0.881358i	$-0.343370\pi$
$348$		0			0
$349$	17.3391			0.928139			0.464069	−	0.885799i	$-0.346389\pi$
$349$	17.3391			0.928139			0.464069	+	0.885799i	$0.346389\pi$
$350$		0			0
$351$			4.50229i			0.240315i
$352$		0			0
$353$	7.90388			0.420681			0.210340	−	0.977628i	$-0.432543\pi$
$353$	7.90388			0.420681			0.210340	+	0.977628i	$0.432543\pi$
$354$		0			0
$355$	2.39857			0.127303
$356$		0			0
$357$	2.51650			0.133187
$358$		0			0
$359$	6.85014			0.361537			0.180768	−	0.983526i	$-0.442142\pi$
$359$	6.85014			0.361537			0.180768	+	0.983526i	$0.442142\pi$
$360$		0			0
$361$	12.0910			0.636368
$362$		0			0
$363$			30.3784i			1.59446i
$364$		0			0
$365$		−	9.23774i		−	0.483526i
$366$		0			0
$367$	−22.4163			−1.17012			−0.585060	−	0.810990i	$-0.698929\pi$
$367$	−22.4163			−1.17012			−0.585060	+	0.810990i	$0.698929\pi$
$368$		0			0
$369$	2.07053			0.107787
$370$		0			0
$371$			4.14832i			0.215370i
$372$		0			0
$373$			31.4598i			1.62893i	0.580214	+	0.814464i	$0.302969\pi$
$373$			31.4598i			1.62893i	−0.580214	+	0.814464i	$0.697031\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	10.3722			0.534195
$378$		0			0
$379$	9.23870			0.474560			0.237280	−	0.971441i	$-0.423744\pi$
$379$	9.23870			0.474560			0.237280	+	0.971441i	$0.423744\pi$
$380$		0			0
$381$	−16.8377			−0.862620
$382$		0			0
$383$	20.2393			1.03418			0.517089	−	0.855932i	$-0.327016\pi$
$383$	20.2393			1.03418			0.517089	+	0.855932i	$0.327016\pi$
$384$		0			0
$385$			9.79702i			0.499302i
$386$		0			0
$387$	4.75591			0.241757
$388$		0			0
$389$		−	15.0549i		−	0.763316i	−0.924304	−	0.381658i	$-0.875353\pi$
$389$		−	15.0549i		−	0.763316i	0.924304	−	0.381658i	$-0.124647\pi$
$390$		0			0
$391$	6.19378	+	4.94262i	0.313233	+	0.249959i
$392$		0			0
$393$	11.1242			0.561143
$394$		0			0
$395$			11.4088i			0.574041i
$396$		0			0
$397$	30.0861			1.50998			0.754989	−	0.655737i	$-0.227642\pi$
$397$	30.0861			1.50998			0.754989	+	0.655737i	$0.227642\pi$
$398$		0			0
$399$		−	8.49227i		−	0.425145i
$400$		0			0
$401$		−	34.8412i		−	1.73989i	−0.493151	−	0.869944i	$-0.664155\pi$
$401$		−	34.8412i		−	1.73989i	0.493151	−	0.869944i	$-0.335845\pi$
$402$		0			0
$403$			7.98216i			0.397620i
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$		−	34.9324i		−	1.73153i
$408$		0			0
$409$	25.1224			1.24223			0.621113	−	0.783721i	$-0.286681\pi$
$409$	25.1224			1.24223			0.621113	+	0.783721i	$0.286681\pi$
$410$		0			0
$411$	−5.56165			−0.274336
$412$		0			0
$413$			6.07770i			0.299064i
$414$		0			0
$415$			9.35972i			0.459451i
$416$		0			0
$417$	5.63364			0.275881
$418$		0			0
$419$	−22.1667			−1.08291			−0.541456	−	0.840729i	$-0.682127\pi$
$419$	−22.1667			−1.08291			−0.541456	+	0.840729i	$0.682127\pi$
$420$		0			0
$421$		−	7.99216i		−	0.389514i	−0.980852	−	0.194757i	$-0.937608\pi$
$421$		−	7.99216i		−	0.389514i	0.980852	−	0.194757i	$-0.0623918\pi$
$422$		0			0
$423$			2.99382i			0.145564i
$424$		0			0
$425$		−	1.65230i		−	0.0801485i
$426$		0			0
$427$		−	14.8435i		−	0.718329i
$428$		0			0
$429$			28.9615i			1.39827i
$430$		0			0
$431$	38.2692			1.84336			0.921682	−	0.387946i	$-0.126815\pi$
$431$	38.2692			1.84336			0.921682	+	0.387946i	$0.126815\pi$
$432$		0			0
$433$			22.5291i			1.08268i	0.840805	+	0.541339i	$0.182082\pi$
$433$			22.5291i			1.08268i	−0.840805	+	0.541339i	$0.817918\pi$
$434$		0			0
$435$	2.30375			0.110457
$436$		0			0
$437$	16.6796	−	20.9018i	0.797892	−	0.999868i
$438$		0			0
$439$		−	17.6828i		−	0.843954i	−0.906606	−	0.421977i	$-0.861336\pi$
$439$		−	17.6828i		−	0.843954i	0.906606	−	0.421977i	$-0.138664\pi$
$440$		0			0
$441$	4.68040			0.222876
$442$		0			0
$443$		−	14.2484i		−	0.676962i	−0.940973	−	0.338481i	$-0.890087\pi$
$443$		−	14.2484i		−	0.676962i	0.940973	−	0.338481i	$-0.109913\pi$
$444$		0			0
$445$	2.66224			0.126202
$446$		0			0
$447$	10.5221			0.497676
$448$		0			0
$449$	−23.2577			−1.09760			−0.548800	−	0.835954i	$-0.684915\pi$
$449$	−23.2577			−1.09760			−0.548800	+	0.835954i	$0.684915\pi$
$450$		0			0
$451$	13.3189			0.627162
$452$		0			0
$453$	17.5034			0.822383
$454$		0			0
$455$			6.85710i			0.321466i
$456$		0			0
$457$		−	19.8933i		−	0.930569i	−0.885161	−	0.465285i	$-0.845952\pi$
$457$		−	19.8933i		−	0.930569i	0.885161	−	0.465285i	$-0.154048\pi$
$458$		0			0
$459$	1.65230			0.0771230
$460$		0			0
$461$	5.03647			0.234572			0.117286	−	0.993098i	$-0.462581\pi$
$461$	5.03647			0.234572			0.117286	+	0.993098i	$0.462581\pi$
$462$		0			0
$463$			14.1754i			0.658788i	0.944193	+	0.329394i	$0.106844\pi$
$463$			14.1754i			0.658788i	−0.944193	+	0.329394i	$0.893156\pi$
$464$		0			0
$465$			1.77291i			0.0822167i
$466$		0			0
$467$	11.8246			0.547176			0.273588	−	0.961847i	$-0.411790\pi$
$467$	11.8246			0.547176			0.273588	+	0.961847i	$0.411790\pi$
$468$		0			0
$469$	−5.66020			−0.261364
$470$		0			0
$471$	−7.31873			−0.337229
$472$		0			0
$473$	30.5929			1.40666
$474$		0			0
$475$	−5.57593			−0.255841
$476$		0			0
$477$			2.72374i			0.124711i
$478$		0			0
$479$	−25.6625			−1.17255			−0.586276	−	0.810112i	$-0.699406\pi$
$479$	−25.6625			−1.17255			−0.586276	+	0.810112i	$0.699406\pi$
$480$		0			0
$481$		−	24.4498i		−	1.11481i
$482$		0			0
$483$	−5.70917	−	4.55590i	−0.259776	−	0.207301i
$484$		0			0
$485$	8.66759			0.393575
$486$		0			0
$487$			17.8398i			0.808397i	0.914671	+	0.404199i	$0.132450\pi$
$487$			17.8398i			0.808397i	−0.914671	+	0.404199i	$0.867550\pi$
$488$		0			0
$489$	−9.06133			−0.409768
$490$		0			0
$491$		−	20.0228i		−	0.903616i	−0.892115	−	0.451808i	$-0.850779\pi$
$491$		−	20.0228i		−	0.903616i	0.892115	−	0.451808i	$-0.149221\pi$
$492$		0			0
$493$		−	3.80650i		−	0.171436i
$494$		0			0
$495$			6.43261i			0.289124i
$496$		0			0
$497$			3.65308i			0.163863i
$498$		0			0
$499$		−	24.8428i		−	1.11212i	−0.831144	−	0.556058i	$-0.812313\pi$
$499$		−	24.8428i		−	1.11212i	0.831144	−	0.556058i	$-0.187687\pi$
$500$		0			0
$501$	8.02942			0.358728
$502$		0			0
$503$	−5.67105			−0.252860			−0.126430	−	0.991976i	$-0.540352\pi$
$503$	−5.67105			−0.252860			−0.126430	+	0.991976i	$0.540352\pi$
$504$		0			0
$505$			18.6952i			0.831925i
$506$		0			0
$507$			7.27066i			0.322901i
$508$		0			0
$509$	12.8447			0.569331			0.284666	−	0.958627i	$-0.408117\pi$
$509$	12.8447			0.569331			0.284666	+	0.958627i	$0.408117\pi$
$510$		0			0
$511$	14.0693			0.622389
$512$		0			0
$513$		−	5.57593i		−	0.246183i
$514$		0			0
$515$			7.63952i			0.336637i
$516$		0			0
$517$			19.2581i			0.846968i
$518$		0			0
$519$		−	0.927933i		−	0.0407317i
$520$		0			0
$521$			12.9412i			0.566963i	0.958978	+	0.283481i	$0.0914894\pi$
$521$			12.9412i			0.566963i	−0.958978	+	0.283481i	$0.908511\pi$
$522$		0			0
$523$	−2.93781			−0.128462			−0.0642308	−	0.997935i	$-0.520459\pi$
$523$	−2.93781			−0.128462			−0.0642308	+	0.997935i	$0.520459\pi$
$524$		0			0
$525$			1.52302i			0.0664702i
$526$		0			0
$527$	2.92939			0.127606
$528$		0			0
$529$	−5.10362	−	22.4266i	−0.221897	−	0.975070i
$530$		0			0
$531$			3.99055i			0.173175i
$532$		0			0
$533$	9.32212			0.403786
$534$		0			0
$535$			6.71244i			0.290204i
$536$		0			0
$537$	−25.1922			−1.08712
$538$		0			0
$539$	30.1072			1.29681
$540$		0			0
$541$	29.0680			1.24973			0.624865	−	0.780733i	$-0.285154\pi$
$541$	29.0680			1.24973			0.624865	+	0.780733i	$0.285154\pi$
$542$		0			0
$543$	−1.69810			−0.0728726
$544$		0			0
$545$	2.70608			0.115916
$546$		0			0
$547$			35.0271i			1.49765i	0.662768	+	0.748825i	$0.269382\pi$
$547$			35.0271i			1.49765i	−0.662768	+	0.748825i	$0.730618\pi$
$548$		0			0
$549$		−	9.74609i		−	0.415953i
$550$		0			0
$551$	−12.8456			−0.547240
$552$		0			0
$553$	−17.3759			−0.738900
$554$		0			0
$555$		−	5.43051i		−	0.230512i
$556$		0			0
$557$		−	17.5781i		−	0.744809i	−0.928071	−	0.372404i	$-0.878533\pi$
$557$		−	17.5781i		−	0.744809i	0.928071	−	0.372404i	$-0.121467\pi$
$558$		0			0
$559$	21.4125			0.905653
$560$		0			0
$561$	10.6286			0.448741
$562$		0			0
$563$	−19.7102			−0.830686			−0.415343	−	0.909665i	$-0.636339\pi$
$563$	−19.7102			−0.830686			−0.415343	+	0.909665i	$0.636339\pi$
$564$		0			0
$565$	−0.212216			−0.00892800
$566$		0			0
$567$	−1.52302			−0.0639610
$568$		0			0
$569$		−	25.5038i		−	1.06918i	−0.845113	−	0.534588i	$-0.820467\pi$
$569$		−	25.5038i		−	1.06918i	0.845113	−	0.534588i	$-0.179533\pi$
$570$		0			0
$571$	−44.2420			−1.85147			−0.925735	−	0.378173i	$-0.876552\pi$
$571$	−44.2420			−1.85147			−0.925735	+	0.378173i	$0.876552\pi$
$572$		0			0
$573$			13.0724i			0.546107i
$574$		0			0
$575$	−2.99135	+	3.74857i	−0.124748	+	0.156326i
$576$		0			0
$577$	−30.3335			−1.26280			−0.631400	−	0.775457i	$-0.717519\pi$
$577$	−30.3335			−1.26280			−0.631400	+	0.775457i	$0.717519\pi$
$578$		0			0
$579$			16.0209i			0.665807i
$580$		0			0
$581$	−14.2551			−0.591400
$582$		0			0
$583$			17.5208i			0.725635i
$584$		0			0
$585$			4.50229i			0.186147i
$586$		0			0
$587$		−	42.0137i		−	1.73409i	−0.498230	−	0.867045i	$-0.666016\pi$
$587$		−	42.0137i		−	1.73409i	0.498230	−	0.867045i	$-0.333984\pi$
$588$		0			0
$589$		−	9.88562i		−	0.407330i
$590$		0			0
$591$		−	11.9113i		−	0.489967i
$592$		0			0
$593$	−45.9637			−1.88750			−0.943752	−	0.330654i	$-0.892731\pi$
$593$	−45.9637			−1.88750			−0.943752	+	0.330654i	$0.892731\pi$
$594$		0			0
$595$	2.51650			0.103166
$596$		0			0
$597$		−	7.54875i		−	0.308950i
$598$		0			0
$599$			4.06616i			0.166139i	0.996544	+	0.0830694i	$0.0264723\pi$
$599$			4.06616i			0.166139i	−0.996544	+	0.0830694i	$0.973528\pi$
$600$		0			0
$601$	−27.3871			−1.11714			−0.558571	−	0.829456i	$-0.688650\pi$
$601$	−27.3871			−1.11714			−0.558571	+	0.829456i	$0.688650\pi$
$602$		0			0
$603$	−3.71642			−0.151344
$604$		0			0
$605$			30.3784i			1.23506i
$606$		0			0
$607$		−	24.1569i		−	0.980496i	−0.871583	−	0.490248i	$-0.836906\pi$
$607$		−	24.1569i		−	0.980496i	0.871583	−	0.490248i	$-0.163094\pi$
$608$		0			0
$609$			3.50867i			0.142179i
$610$		0			0
$611$			13.4791i			0.545304i
$612$		0			0
$613$			21.9152i			0.885147i	0.896732	+	0.442574i	$0.145934\pi$
$613$			21.9152i			0.885147i	−0.896732	+	0.442574i	$0.854066\pi$
$614$		0			0
$615$	2.07053			0.0834917
$616$		0			0
$617$		−	6.74213i		−	0.271428i	−0.990748	−	0.135714i	$-0.956667\pi$
$617$		−	6.74213i		−	0.271428i	0.990748	−	0.135714i	$-0.0433328\pi$
$618$		0			0
$619$	−19.3523			−0.777836			−0.388918	−	0.921272i	$-0.627151\pi$
$619$	−19.3523			−0.777836			−0.388918	+	0.921272i	$0.627151\pi$
$620$		0			0
$621$	−3.74857	−	2.99135i	−0.150425	−	0.120039i
$622$		0			0
$623$			4.05466i			0.162446i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	35.8678i		−	1.43242i
$628$		0			0
$629$	−8.97286			−0.357771
$630$		0			0
$631$	−16.1921			−0.644598			−0.322299	−	0.946638i	$-0.604456\pi$
$631$	−16.1921			−0.644598			−0.322299	+	0.946638i	$0.604456\pi$
$632$		0			0
$633$	−4.69539			−0.186625
$634$		0			0
$635$	−16.8377			−0.668183
$636$		0			0
$637$	21.0725			0.834924
$638$		0			0
$639$			2.39857i			0.0948859i
$640$		0			0
$641$		−	25.9194i		−	1.02376i	−0.859058	−	0.511878i	$-0.828950\pi$
$641$		−	25.9194i		−	1.02376i	0.859058	−	0.511878i	$-0.171050\pi$
$642$		0			0
$643$	23.0513			0.909055			0.454527	−	0.890733i	$-0.349808\pi$
$643$	23.0513			0.909055			0.454527	+	0.890733i	$0.349808\pi$
$644$		0			0
$645$	4.75591			0.187264
$646$		0			0
$647$			38.7579i			1.52373i	0.647735	+	0.761866i	$0.275716\pi$
$647$			38.7579i			1.52373i	−0.647735	+	0.761866i	$0.724284\pi$
$648$		0			0
$649$			25.6696i			1.00762i
$650$		0			0
$651$	−2.70018			−0.105829
$652$		0			0
$653$	−2.54972			−0.0997781			−0.0498891	−	0.998755i	$-0.515887\pi$
$653$	−2.54972			−0.0997781			−0.0498891	+	0.998755i	$0.515887\pi$
$654$		0			0
$655$	11.1242			0.434660
$656$		0			0
$657$	9.23774			0.360399
$658$		0			0
$659$	−43.3121			−1.68720			−0.843600	−	0.536972i	$-0.819568\pi$
$659$	−43.3121			−1.68720			−0.843600	+	0.536972i	$0.819568\pi$
$660$		0			0
$661$		−	9.39709i		−	0.365504i	−0.983159	−	0.182752i	$-0.941499\pi$
$661$		−	9.39709i		−	0.365504i	0.983159	−	0.182752i	$-0.0585006\pi$
$662$		0			0
$663$	7.43916			0.288913
$664$		0			0
$665$		−	8.49227i		−	0.329316i
$666$		0			0
$667$	−6.89134	+	8.63579i	−0.266834	+	0.334379i
$668$		0			0
$669$	24.1553			0.933900
$670$		0			0
$671$		−	62.6928i		−	2.42023i
$672$		0			0
$673$	−2.72747			−0.105136			−0.0525681	−	0.998617i	$-0.516741\pi$
$673$	−2.72747			−0.105136			−0.0525681	+	0.998617i	$0.516741\pi$
$674$		0			0
$675$			1.00000i			0.0384900i
$676$		0			0
$677$		−	12.1061i		−	0.465276i	−0.972563	−	0.232638i	$-0.925264\pi$
$677$		−	12.1061i		−	0.465276i	0.972563	−	0.232638i	$-0.0747358\pi$
$678$		0			0
$679$			13.2009i			0.506606i
$680$		0			0
$681$			12.4903i			0.478629i
$682$		0			0
$683$			20.7694i			0.794720i	0.917663	+	0.397360i	$0.130074\pi$
$683$			20.7694i			0.794720i	−0.917663	+	0.397360i	$0.869926\pi$
$684$		0			0
$685$	−5.56165			−0.212500
$686$		0			0
$687$	−29.8509			−1.13888
$688$		0			0
$689$			12.2631i			0.467186i
$690$		0			0
$691$		−	6.08978i		−	0.231666i	−0.993269	−	0.115833i	$-0.963046\pi$
$691$		−	6.08978i		−	0.231666i	0.993269	−	0.115833i	$-0.0369538\pi$
$692$		0			0
$693$	−9.79702			−0.372158
$694$		0			0
$695$	5.63364			0.213696
$696$		0			0
$697$		−	3.42114i		−	0.129585i
$698$		0			0
$699$		−	0.897082i		−	0.0339308i
$700$		0			0
$701$		−	10.6195i		−	0.401094i	−0.979684	−	0.200547i	$-0.935728\pi$
$701$		−	10.6195i		−	0.401094i	0.979684	−	0.200547i	$-0.0642720\pi$
$702$		0			0
$703$			30.2802i			1.14204i
$704$		0			0
$705$			2.99382i			0.112754i
$706$		0			0
$707$	−28.4732			−1.07085
$708$		0			0
$709$		−	24.7363i		−	0.928991i	−0.885575	−	0.464496i	$-0.846236\pi$
$709$		−	24.7363i		−	0.928991i	0.885575	−	0.464496i	$-0.153764\pi$
$710$		0			0
$711$	−11.4088			−0.427865
$712$		0			0
$713$	−6.64588	−	5.30340i	−0.248890	−	0.198614i
$714$		0			0
$715$			28.9615i			1.08310i
$716$		0			0
$717$	25.9763			0.970101
$718$		0			0
$719$		−	51.3791i		−	1.91612i	−0.286571	−	0.958059i	$-0.592515\pi$
$719$		−	51.3791i		−	1.91612i	0.286571	−	0.958059i	$-0.407485\pi$
$720$		0			0
$721$	−11.6352			−0.433316
$722$		0			0
$723$	8.15296			0.303212
$724$		0			0
$725$	2.30375			0.0855593
$726$		0			0
$727$	−5.94441			−0.220466			−0.110233	−	0.993906i	$-0.535160\pi$
$727$	−5.94441			−0.220466			−0.110233	+	0.993906i	$0.535160\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	7.85822i		−	0.290647i
$732$		0			0
$733$			47.7046i			1.76201i	0.473106	+	0.881005i	$0.343133\pi$
$733$			47.7046i			1.76201i	−0.473106	+	0.881005i	$0.656867\pi$
$734$		0			0
$735$	4.68040			0.172639
$736$		0			0
$737$	−23.9063			−0.880599
$738$		0			0
$739$			30.6154i			1.12621i	0.826387	+	0.563103i	$0.190393\pi$
$739$			30.6154i			1.12621i	−0.826387	+	0.563103i	$0.809607\pi$
$740$		0			0
$741$		−	25.1045i		−	0.922236i
$742$		0			0
$743$	−4.15462			−0.152418			−0.0762092	−	0.997092i	$-0.524282\pi$
$743$	−4.15462			−0.152418			−0.0762092	+	0.997092i	$0.524282\pi$
$744$		0			0
$745$	10.5221			0.385498
$746$		0			0
$747$	−9.35972			−0.342454
$748$		0			0
$749$	−10.2232			−0.373548
$750$		0			0
$751$	4.49223			0.163924			0.0819619	−	0.996635i	$-0.473881\pi$
$751$	4.49223			0.163924			0.0819619	+	0.996635i	$0.473881\pi$
$752$		0			0
$753$		−	6.31194i		−	0.230020i
$754$		0			0
$755$	17.5034			0.637015
$756$		0			0
$757$		−	7.70763i		−	0.280138i	−0.990142	−	0.140069i	$-0.955267\pi$
$757$		−	7.70763i		−	0.280138i	0.990142	−	0.140069i	$-0.0447325\pi$
$758$		0			0
$759$	−24.1131	−	19.2422i	−0.875250	−	0.698448i
$760$		0			0
$761$	−2.85989			−0.103671			−0.0518355	−	0.998656i	$-0.516507\pi$
$761$	−2.85989			−0.103671			−0.0518355	+	0.998656i	$0.516507\pi$
$762$		0			0
$763$			4.12143i			0.149206i
$764$		0			0
$765$	1.65230			0.0597392
$766$		0			0
$767$			17.9666i			0.648737i
$768$		0			0
$769$		−	0.422431i		−	0.0152333i	−0.999971	−	0.00761663i	$-0.997576\pi$
$769$		−	0.422431i		−	0.0152333i	0.999971	−	0.00761663i	$-0.00242447\pi$
$770$		0			0
$771$		−	10.8718i		−	0.391538i
$772$		0			0
$773$		−	13.0769i		−	0.470342i	−0.971954	−	0.235171i	$-0.924435\pi$
$773$		−	13.0769i		−	0.470342i	0.971954	−	0.235171i	$-0.0755650\pi$
$774$		0			0
$775$			1.77291i			0.0636848i
$776$		0			0
$777$	8.27080			0.296713
$778$		0			0
$779$	−11.5451			−0.413647
$780$		0			0
$781$			15.4291i			0.552095i
$782$		0			0
$783$			2.30375i			0.0823295i
$784$		0			0
$785$	−7.31873			−0.261217
$786$		0			0
$787$	44.8315			1.59807			0.799035	−	0.601285i	$-0.205344\pi$
$787$	44.8315			1.59807			0.799035	+	0.601285i	$0.205344\pi$
$788$		0			0
$789$			23.5415i			0.838098i
$790$		0			0
$791$		−	0.323210i		−	0.0114920i
$792$		0			0
$793$		−	43.8798i		−	1.55822i
$794$		0			0
$795$			2.72374i			0.0966011i
$796$		0			0
$797$			27.9408i			0.989715i	0.868974	+	0.494858i	$0.164780\pi$
$797$			27.9408i			0.989715i	−0.868974	+	0.494858i	$0.835220\pi$
$798$		0			0
$799$	4.94670			0.175002
$800$		0			0
$801$			2.66224i			0.0940657i
$802$		0			0
$803$	59.4228			2.09698
$804$		0			0
$805$	−5.70917	−	4.55590i	−0.201222	−	0.160574i
$806$		0			0
$807$			13.5049i			0.475395i
$808$		0			0
$809$	−25.4099			−0.893364			−0.446682	−	0.894693i	$-0.647394\pi$
$809$	−25.4099			−0.893364			−0.446682	+	0.894693i	$0.647394\pi$
$810$		0			0
$811$			3.40842i			0.119686i	0.998208	+	0.0598430i	$0.0190600\pi$
$811$			3.40842i			0.119686i	−0.998208	+	0.0598430i	$0.980940\pi$
$812$		0			0
$813$	−18.2034			−0.638421
$814$		0			0
$815$	−9.06133			−0.317405
$816$		0			0
$817$	−26.5186			−0.927770
$818$		0			0
$819$	−6.85710			−0.239607
$820$		0			0
$821$	35.3534			1.23384			0.616922	−	0.787024i	$-0.288379\pi$
$821$	35.3534			1.23384			0.616922	+	0.787024i	$0.288379\pi$
$822$		0			0
$823$			40.5793i			1.41450i	0.706961	+	0.707252i	$0.250066\pi$
$823$			40.5793i			1.41450i	−0.706961	+	0.707252i	$0.749934\pi$
$824$		0			0
$825$			6.43261i			0.223955i
$826$		0			0
$827$	−12.0863			−0.420281			−0.210141	−	0.977671i	$-0.567392\pi$
$827$	−12.0863			−0.420281			−0.210141	+	0.977671i	$0.567392\pi$
$828$		0			0
$829$	24.2928			0.843724			0.421862	−	0.906660i	$-0.361377\pi$
$829$	24.2928			0.843724			0.421862	+	0.906660i	$0.361377\pi$
$830$		0			0
$831$		−	9.79569i		−	0.339809i
$832$		0			0
$833$		−	7.73344i		−	0.267948i
$834$		0			0
$835$	8.02942			0.277869
$836$		0			0
$837$	−1.77291			−0.0612807
$838$		0			0
$839$	28.0086			0.966963			0.483482	−	0.875354i	$-0.339372\pi$
$839$	28.0086			0.966963			0.483482	+	0.875354i	$0.339372\pi$
$840$		0			0
$841$	−23.6927			−0.816990
$842$		0			0
$843$	10.6935			0.368305
$844$		0			0
$845$			7.27066i			0.250118i
$846$		0			0
$847$	−46.2671			−1.58976
$848$		0			0
$849$		−	27.2186i		−	0.934138i
$850$		0			0
$851$	20.3567	+	16.2446i	0.697818	+	0.556857i
$852$		0			0
$853$	−3.61457			−0.123761			−0.0618803	−	0.998084i	$-0.519710\pi$
$853$	−3.61457			−0.123761			−0.0618803	+	0.998084i	$0.519710\pi$
$854$		0			0
$855$		−	5.57593i		−	0.190693i
$856$		0			0
$857$	−16.7340			−0.571622			−0.285811	−	0.958286i	$-0.592263\pi$
$857$	−16.7340			−0.571622			−0.285811	+	0.958286i	$0.592263\pi$
$858$		0			0
$859$			15.4101i			0.525786i	0.964825	+	0.262893i	$0.0846767\pi$
$859$			15.4101i			0.525786i	−0.964825	+	0.262893i	$0.915323\pi$
$860$		0			0
$861$			3.15346i			0.107470i
$862$		0			0
$863$			15.7961i			0.537706i	0.963181	+	0.268853i	$0.0866446\pi$
$863$			15.7961i			0.537706i	−0.963181	+	0.268853i	$0.913355\pi$
$864$		0			0
$865$		−	0.927933i		−	0.0315507i
$866$		0			0
$867$			14.2699i			0.484631i
$868$		0			0
$869$	−73.3887			−2.48954
$870$		0			0
$871$	−16.7324			−0.566956
$872$		0			0
$873$			8.66759i			0.293353i
$874$		0			0
$875$			1.52302i			0.0514876i
$876$		0			0
$877$	40.5977			1.37089			0.685444	−	0.728125i	$-0.259608\pi$
$877$	40.5977			1.37089			0.685444	+	0.728125i	$0.259608\pi$
$878$		0			0
$879$	−23.1769			−0.781737
$880$		0			0
$881$		−	51.3114i		−	1.72873i	−0.502869	−	0.864363i	$-0.667722\pi$
$881$		−	51.3114i		−	1.72873i	0.502869	−	0.864363i	$-0.332278\pi$
$882$		0			0
$883$		−	11.1535i		−	0.375346i	−0.982232	−	0.187673i	$-0.939906\pi$
$883$		−	11.1535i		−	0.375346i	0.982232	−	0.187673i	$-0.0600945\pi$
$884$		0			0
$885$			3.99055i			0.134141i
$886$		0			0
$887$		−	8.68530i		−	0.291624i	−0.989312	−	0.145812i	$-0.953421\pi$
$887$		−	8.68530i		−	0.291624i	0.989312	−	0.145812i	$-0.0465794\pi$
$888$		0			0
$889$		−	25.6442i		−	0.860078i
$890$		0			0
$891$	−6.43261			−0.215500
$892$		0			0
$893$		−	16.6933i		−	0.558621i
$894$		0			0
$895$	−25.1922			−0.842081
$896$		0			0
$897$	−16.8772	−	13.4680i	−0.563513	−	0.449682i
$898$		0			0
$899$			4.08435i			0.136221i
$900$		0			0
$901$	4.50045			0.149932
$902$		0			0
$903$			7.24337i			0.241044i
$904$		0			0
$905$	−1.69810			−0.0564468
$906$		0			0
$907$	39.9169			1.32542			0.662709	−	0.748877i	$-0.269407\pi$
$907$	39.9169			1.32542			0.662709	+	0.748877i	$0.269407\pi$
$908$		0			0
$909$	−18.6952			−0.620081
$910$		0			0
$911$	5.43948			0.180218			0.0901091	−	0.995932i	$-0.471278\pi$
$911$	5.43948			0.180218			0.0901091	+	0.995932i	$0.471278\pi$
$912$		0			0
$913$	−60.2074			−1.99257
$914$		0			0
$915$		−	9.74609i		−	0.322196i
$916$		0			0
$917$			16.9425i			0.559490i
$918$		0			0
$919$	33.6490			1.10998			0.554989	−	0.831858i	$-0.312723\pi$
$919$	33.6490			1.10998			0.554989	+	0.831858i	$0.312723\pi$
$920$		0			0
$921$	23.1278			0.762086
$922$		0			0
$923$			10.7991i			0.355455i
$924$		0			0
$925$		−	5.43051i		−	0.178554i
$926$		0			0
$927$	−7.63952			−0.250915
$928$		0			0
$929$	56.3592			1.84909			0.924543	−	0.381077i	$-0.124447\pi$
$929$	56.3592			1.84909			0.924543	+	0.381077i	$0.124447\pi$
$930$		0			0
$931$	−26.0976			−0.855313
$932$		0			0
$933$	15.5382			0.508698
$934$		0			0
$935$	10.6286			0.347593
$936$		0			0
$937$			0.622601i			0.0203395i	0.999948	+	0.0101697i	$0.00323718\pi$
$937$			0.622601i			0.0203395i	−0.999948	+	0.0101697i	$0.996763\pi$
$938$		0			0
$939$	1.93700			0.0632116
$940$		0			0
$941$		−	17.7044i		−	0.577147i	−0.957458	−	0.288574i	$-0.906819\pi$
$941$		−	17.7044i		−	0.577147i	0.957458	−	0.288574i	$-0.0931810\pi$
$942$		0			0
$943$	−6.19368	+	7.76153i	−0.201694	+	0.252750i
$944$		0			0
$945$	−1.52302			−0.0495440
$946$		0			0
$947$			0.0434608i			0.00141229i	1.00000		0.000706143i	$0.000224772\pi$
$947$			0.0434608i			0.00141229i	−1.00000		0.000706143i	$0.999775\pi$
$948$		0			0
$949$	41.5910			1.35010
$950$		0			0
$951$		−	20.0281i		−	0.649457i
$952$		0			0
$953$		−	54.2160i		−	1.75623i	−0.478452	−	0.878114i	$-0.658802\pi$
$953$		−	54.2160i		−	1.75623i	0.478452	−	0.878114i	$-0.341198\pi$
$954$		0			0
$955$			13.0724i			0.423013i
$956$		0			0
$957$			14.8191i			0.479035i
$958$		0			0
$959$		−	8.47052i		−	0.273527i
$960$		0			0
$961$	27.8568			0.898606
$962$		0			0
$963$	−6.71244			−0.216305
$964$		0			0
$965$			16.0209i			0.515732i
$966$		0			0
$967$		−	8.23566i		−	0.264841i	−0.991194	−	0.132420i	$-0.957725\pi$
$967$		−	8.23566i		−	0.264841i	0.991194	−	0.132420i	$-0.0422749\pi$
$968$		0			0
$969$	−9.21313			−0.295968
$970$		0			0
$971$	−9.55880			−0.306756			−0.153378	−	0.988168i	$-0.549015\pi$
$971$	−9.55880			−0.306756			−0.153378	+	0.988168i	$0.549015\pi$
$972$		0			0
$973$			8.58017i			0.275068i
$974$		0			0
$975$			4.50229i			0.144189i
$976$		0			0
$977$		−	50.3198i		−	1.60987i	−0.593361	−	0.804936i	$-0.702199\pi$
$977$		−	50.3198i		−	1.60987i	0.593361	−	0.804936i	$-0.297801\pi$
$978$		0			0
$979$			17.1252i			0.547322i
$980$		0			0
$981$			2.70608i			0.0863986i
$982$		0			0
$983$	−18.7966			−0.599519			−0.299760	−	0.954015i	$-0.596906\pi$
$983$	−18.7966			−0.599519			−0.299760	+	0.954015i	$0.596906\pi$
$984$		0			0
$985$		−	11.9113i		−	0.379526i
$986$		0			0
$987$	−4.55966			−0.145135
$988$		0			0
$989$	−14.2266	+	17.8279i	−0.452380	+	0.566894i
$990$		0			0
$991$		−	32.1729i		−	1.02200i	−0.859579	−	0.511002i	$-0.829274\pi$
$991$		−	32.1729i		−	1.02200i	0.859579	−	0.511002i	$-0.170726\pi$
$992$		0			0
$993$	19.6804			0.624538
$994$		0			0
$995$		−	7.54875i		−	0.239311i
$996$		0			0
$997$	−4.15041			−0.131445			−0.0657224	−	0.997838i	$-0.520935\pi$
$997$	−4.15041			−0.131445			−0.0657224	+	0.997838i	$0.520935\pi$
$998$		0			0
$999$	5.43051			0.171814

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.be.c.1471.4	yes	32
4.3	odd	2		5520.2.be.d.1471.3	yes	32
23.22	odd	2		5520.2.be.d.1471.4	yes	32
92.91	even	2	inner	5520.2.be.c.1471.3	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5520.2.be.c.1471.3	✓	32	92.91	even	2	inner
5520.2.be.c.1471.4	yes	32	1.1	even	1	trivial
5520.2.be.d.1471.3	yes	32	4.3	odd	2
5520.2.be.d.1471.4	yes	32	23.22	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.be (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000i q^{5} -1.52302 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +1.00000i q^{5} -1.52302 q^{7} -1.00000 q^{9} -6.43261 q^{11} -4.50229 q^{13} -1.00000 q^{15} +1.65230i q^{17} +5.57593 q^{19} -1.52302i q^{21} +(2.99135 - 3.74857i) q^{23} -1.00000 q^{25} -1.00000i q^{27} -2.30375 q^{29} -1.77291i q^{31} -6.43261i q^{33} -1.52302i q^{35} +5.43051i q^{37} -4.50229i q^{39} -2.07053 q^{41} -4.75591 q^{43} -1.00000i q^{45} -2.99382i q^{47} -4.68040 q^{49} -1.65230 q^{51} -2.72374i q^{53} -6.43261i q^{55} +5.57593i q^{57} -3.99055i q^{59} +9.74609i q^{61} +1.52302 q^{63} -4.50229i q^{65} +3.71642 q^{67} +(3.74857 + 2.99135i) q^{69} -2.39857i q^{71} -9.23774 q^{73} -1.00000i q^{75} +9.79702 q^{77} +11.4088 q^{79} +1.00000 q^{81} +9.35972 q^{83} -1.65230 q^{85} -2.30375i q^{87} -2.66224i q^{89} +6.85710 q^{91} +1.77291 q^{93} +5.57593i q^{95} -8.66759i q^{97} +6.43261 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7} - 32 q^{9}+O(q^{10}) \) 32 * q - 8 * q^7 - 32 * q^9	\( 32 q - 8 q^{7} - 32 q^{9} + 8 q^{11} - 8 q^{13} - 32 q^{15} - 32 q^{25} + 4 q^{29} + 20 q^{41} + 52 q^{49} - 4 q^{51} + 8 q^{63} + 32 q^{67} - 40 q^{73} - 24 q^{77} + 32 q^{79} + 32 q^{81} - 4 q^{85} - 48 q^{91} - 8 q^{99}+O(q^{100}) \) 32 * q - 8 * q^7 - 32 * q^9 + 8 * q^11 - 8 * q^13 - 32 * q^15 - 32 * q^25 + 4 * q^29 + 20 * q^41 + 52 * q^49 - 4 * q^51 + 8 * q^63 + 32 * q^67 - 40 * q^73 - 24 * q^77 + 32 * q^79 + 32 * q^81 - 4 * q^85 - 48 * q^91 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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